Problem: $ A = \left[\begin{array}{rr}3 & 5 \\ 3 & -2 \\ -1 & -2\end{array}\right]$ $ B = \left[\begin{array}{rr}-1 & 2 \\ 5 & 2\end{array}\right]$ What is $ A B$ ?
Answer: Because $ A$ has dimensions $(3\times2)$ and $ B$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ A B = \left[\begin{array}{rr}{3} & {5} \\ {3} & {-2} \\ \color{gray}{-1} & \color{gray}{-2}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{2} \\ {5} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5} & ? \\ {3}\cdot{-1}+{-2}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5} & {3}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{2} \\ {3}\cdot{-1}+{-2}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{-1}+{5}\cdot{5} & {3}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{2} \\ {3}\cdot{-1}+{-2}\cdot{5} & {3}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{2} \\ \color{gray}{-1}\cdot{-1}+\color{gray}{-2}\cdot{5} & \color{gray}{-1}\cdot\color{#DF0030}{2}+\color{gray}{-2}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}22 & 16 \\ -13 & 2 \\ -9 & -6\end{array}\right] $